Poincaré's reversibility condition (Q5945757)

Consider a real planar analytic vector field where the origin is a center for the linearization of the planar field. The Poincaré reversibility condition says that if the system is reversible with respect to a line which passes through the origin, then the origin is a center. The author gives necessary and sufficient conditions for the existence of a reversibility line for any analytic system. Quadratic and cubic systems as well as non-Hamiltonian systems are studied.